1. Calculation of the DFT

Transcription

1 ELE E4810: Digital Signal Processing Topic 10: The Fast Fourier Transform 1. Calculation of the DFT. The Fast Fourier Transform algorithm 3. Short-Time Fourier Transform 1 1. Calculation of the DFT! Filter design so far has been oriented to time-domain processing - cheaper!! But: frequency-domain processing makes some problems very simple: x[n] DFT X[k] Fourier domain processing Y[k] IDFT y[n]! eed an efficient way to calculate DFT!

2 The DFT! Recall the DFT: X[k] = "1 # n=0! discrete transform of discrete sequence! Matrix form: x[n]w kn # X[0] & # L 1 &# x[0] & % ( % 1 ("1) (% ( % X[1] ( % 1 W W L W (% x[1] ( 4 ("1) % X[] ( = % 1 W W L W (% x[] ( % M ( % M M M O M (% M ( % ( % (% ( ("1) ("1) ("1) $ X[ "1] ' $ 1 W W L W ' $ x[ "1] ' ( W = e " j# ) " W r has only distinct values Lots of structure $ opportunities for efficient algorithms 3 Computational Complexity X[k] = "1 # n=0 x[n]w kn! cpx multiplies + -1 cpx adds per pt! points (k = 0..-1)! cpx mult: (a+jb)(c+jd) = ac - bd + j(ad+bc) = 4 real mults + real adds! cpx add = real adds! Total: 4 real mults, 4 - real adds 4

3 Goertzel s Algorithm "1 kl! ow: X[ k] = # x[ l]w! i.e. x e [n] x e [] = 0 where + W -k X k n y k z -1 l=0 k "k "l = W # x[ l] W l [ ] = y k [ ] [ ] = x e n [ ]" h k [n] y k [n] y k [-1] = 0 y k [] = X[k] ( ) looks like a convolution x e [n] = { x[n] 0! n < 0 n = h k [n] = { W -kn n " 0 0 n < 0 5 H( z) = Goertzel s Algorithm! Separate filters for each X[k]! can calculate for just a few samples of k! o large buffer, no coefficient table! Same complexity for full X[k] (4 mults, 4 - adds)! but: can halve multiplies by making the denominator real: k z "1 1 1" W "k z = 1" W "1 1 " cos #k z"1 + z " evaluate only for last step 6 real mults per step

4 . Fast Fourier Transform FFT! Reduce complexity of DFT from O( ) to O( log)! grows more slowly with larger! Works by decomposing large DFT into several stages of smaller DFTs! Often provided as a highly optimized library 7 Arrange terms in pairs Decimation in Time (DIT) FFT! Can rearrange DFT formula in halves: nk [ ] = x[ n] "W X k Group terms from each pair #1 $ n=0 #1 $ [ ] " W mk ( m+1)k ( + x[ m +1] " W ) = x m m=0 #1 mk k = $ x m + W [ ] "W #1 $ [ ] "W x m +1 m=0 X m=0 0 [<k> / ] X 1 [<k> / ] / pt DFT of x for even n / pt DFT of x for odd n 8 mk

5 X 0 z Decimation in Time (DIT) FFT! Finite sequence x[n], 0! n <, = M! i.e. length is a power of! Divide z-transform into parts coming from even and odd values of n: "1 X( z) = # x[ n]z "n = X 0 z n=0 # "1 [ ]z "n X 1 z ( ) = x n n=0 ZT of / point sequence formed from even pts of x[n] ( ) + z "1 X 1 ( z ) ( ) = # "1 x n +1 n=0 ZT of / point sequence formed from odd pts of x[n] [ ]z "n 9 Decimation in Time (DIT) FFT! ow, but = X 0 e! Hence: X k k = DFT { x[ n] } = ˆ X[ k] = X( z) z=e j "k / #1 =W j "k / ( ) ( ) + e# j "k / X ( j "k / 1 e ) ( e j"k / ) j"k / / = e [ ] = DFT = X [ 0 k ] + W ( ) = W k X 1 [ ] k ( ) #1 { x[ n] }+ W k DFT { x[ n +1] } / point DFT defined only for k = 0../-1 10

6 Decimation in Time (DIT) FFT DFT { x[ n] } = DFT { x 0 [ n] }+ W k DFT { x 1 [ n] }! We can evaluate an -pt DFT as two /-pt DFTs (plus a few mults/adds)! But if DFT { } ~ O( ) then DFT / { } ~ O((/) ) = 1/4 O( ) " Total computation ~! 1/4 O( ) = 1/ the computation (+%) of direct DFT 11 One-Stage DIT Flowgraph [ ] = X 0 k X k Even points from x[n] [ ] + W k X 1 [ ] k twiddle factors always apply to odd-terms ouptut OT mirror-image Odd points from x[n] Classic FFT structure Same as X[0..3] except for factors on X 1 [ ] terms 1

7 Multiple DIT Stages! If decomposing one DFT into two smaller DFT / s speeds things up... Why not further divide into DFT /4 s? i.e. so have! Similarly, [ ] = X 0 k X k 0! k < X 0 k [ ] + W [ ] = X 00 k 4 0! k < / /4-pt DFT of even points in even subset of x[n] X 1 k k X 1 [ ] k [ ] + W [ ] = X 10 k 4 13 k X 01 [ ] k 4 /4-pt DFT of odd points from even subset [ ] + W k X 11 [ ] k 4 Two-Stage DIT Flowgraph 14

8 Multi-stage DIT FFT! Can keep doing this until we get down to -pt DFTs: DFT X[0] = x[0] + x[1] X[1] = x[0] - x[1] butterfly element 1 = W 0-1 = W 1 $ = M -pt DFT reduces to M stages of twiddle factors & summation (O( ) part vanishes) $ real mults < M 4, real adds <!M $ complexity ~ O( M) = O( log ) 15 & FFT Implementation Details! Basic butterfly (at any stage): X X0 [r] X X1 [r]! Can simplify: X X0 [r] X X1 [r] W r -1 W r W r+/ X X [r] cpx mults X X [r+/] X X [r] X X [r+/] W r+ ( ) = j # r+ e" = e " j#r / " j# $ e r = "W just one cpx mult! i.e. SUB rather than ADD 16

9 bit-reversed indexing 8-pt DIT FFT Flowgraph! -1 s absorbed into summation nodes! W 0 disappears! in-place algorithm: sequential stages 17 FFT for Other Values of! Having = M meant we could divide each stage into halves = radix- FFT! Same approach works for:! = 3 M radix-3! = 4 M radix-4 - more optimized radix-! etc...! Composite = a b c d $ mixed radix (different /r point FFTs at each stage)!.. or just zero-pad to make = M 18

10 Inverse FFT! Recall IDFT: x[n] = 1 "1 # k=0 only differences from forward DFT X[k]W "nk x n! Thus: x * [n] = "1 "nk #( X[k]W ) * = X * nk # [k]w k=0 Forward DFT of x'[n] = X * [k] k=n i.e. time sequence made from spectrum "1 k=0! Hence, use FFT to calculate IFFT: $ & % "1 # [ ] = 1 X * k k=0 ' nk [ ]W ) ( * Re{X[k]} Im{X[k]} DFT 19-1 Re Im Re Im 1/ -1/ Re{x[n]} Im{x[n]} DFT of Real Sequences! If x[n] is pure-real, DFT wastes mult s! Real x[n] $ Conj. symm. X[k] = X * [-k]! Given two real sequences, x[n] and w[n] call y[n] = j w[n], v[n] = x[n] + y[n]! -pt DFT V[k] = X[k] + Y[k] but: V[k]+V * [-k] = X[k]+X * X[k] -Y[k] [-k]+y[k]+y * [-k] " X[k]= 1 / (V[k]+V * [-k]), W[k]= -j / (V[k]-V * [-k])! i.e. compute DFTs of two -pt real sequences with a single -pt DFT 0 M

11 x n 3. Short-Time Fourier Transform (STFT)! Fourier Transform (e.g. DTFT) gives spectrum of an entire sequence:! How to see a time-varying spectrum?! e.g. slow AM of a sinusoid carrier: $ % [ ] = & 1 " cos #n ' ) cos* ( 0 n 1 Fourier Transform of AM Sine! Spectrum of whole sequence indicates modulation indirectly...!... as cancellation between closelytuned sines sin#kn -sin#(k-1)n -sin#(k+1)n cacb = ca+b +ca-b

12 Fourier Transform of AM Sine! Sometimes we d rather separate modulation and carrier: x[n] = A[n]cos' 0 n! A[n] varies on a different (slower) timescale! One approach:! chop x[n] into short sub-sequences ' 0 A[n] '!.. each with ~ constant modulation scale! DFT spectrum of pieces $ show variation 3 FT of Short Segments! Break up x[n] into successive, shorter chunks of length FT, then DFT each: Shows amplitude modulation of ' 0 energy 4

13 The Spectrogram! Plot successive DFTs in time-frequency: time hopsize (between successive frames) = 18 points! This image is called the Spectrogram 5 Short-Time Fourier Transform! Spectrogram = STFT magnitude plotted on time-frequency plane! STFT is (DFT form): frequency index [ ] = x[ n 0 + n] " w[n]" e # j $kn FT X k,n 0 time index FT #1 % n=0 FT points of x starting at n window DFT kernel! intensity as a function of time & frequency 6

14 DTFT form of STFT STFT Window Shape! w[n] provides time localization of STFT w[n]! e.g. rectangular selects x[n], n 0! n < n 0 + W! But: resulting spectrum has same problems as windowing for FIR design: X( e j",n 0 ) = DTFT{ x[ n 0 + n] # w[ n] } & = e j$n 0 ' X ( e j$ )W ( e j ("%$ ) )d$ %& spectrum of short-time window is convolved with (twisted) parent spectrum n 7 STFT Window Shape! e.g. if x[n] is a pure sinusoid, blurring (mainlobe) + ghosting (sidelobes)! Hence, use tapered window for w[n] e.g. Hamming w[ n] = cos(" M n ) +1 sidelobes < -40 db 8

15 STFT Window Length! Length of w[n] sets temporal resolution short window measures only local properties longer window averages spectral character! Window length ( 1/(Mainlobe width) shorter window $ more blurred spectrum! more time detail ) less frequency detail 9 STFT Window Length! Can illustrate time-frequency tradeoff on the time-frequency plane:! Alternate tilings of time-freq: disks show blurring due to window length area of disks is constant $ Uncertainty principle: *f *t! k half-length window $ half as many DFT samples 30

16 Spectrograms of Real Sounds time-domain successive short DFTs individual t-f cells merge into continuous image 31 arrowband vs. Wideband! Effect of varying window length: 3

17 Spectrogram in Matlab >> [d,sr]=wavread('sx419.wav'); >> w=56; >> specgram(d,w,sr) >> colorbar (hann) window length actual sampling rate (to label time axis) db 33 just one freq. STFT as a Filterbank! Consider one row of STFT: [ ] = x[ n 0 + n] " w[ n] " e # j $kn X k n 0 where #1 % n=0 # #1 ( ) % = h k [ m]x n 0 # m m=0 h k n [ ] [ ] = w ["n] # e j $kn convolution with complex IR! Each STFT row is output of a filter (subsampled by the STFT hop size) 34

18 STFT as a Filterbank! If h k [ n] = w [(")n] # e j $kn then ( ) = W e # H k e j" ( $k ( ) j ("# )) shift-in-'! Each STFT row is the same bandpass response defined by W(e j' ), frequency-shifted to a given DFT bin: A bank of identical, frequency-shifted bandpass filters: filterbank 35 STFT Analysis-Synthesis! IDFT of STFT frames can reconstruct (part of) original waveform [ ] = DFT{ x[ n 0 + n] " w[ n] }! e.g. if X k,n 0 then IDFT{ X[ k,n 0 ]} = x[ n 0 + n] " w[ n]! Can shift by n 0, combine, to get x[n]: ^! Could divide by w[n-n 0 ] to recover x[n]... 36

19 STFT Analysis-Synthesis! Dividing by small values of w[n] is bad! Prefer to overlap windows: i.e. sample X[k,n 0 ] at n 0 = r H where H = / (for example)! Then x ˆ n hopsize # r window length [ ] = x[ n]w[ n " rh] = x[ n] if w[ n " rh] $ =1 37 #r STFT Analysis-Synthesis! Hann or Hamming windows with 50% overlap sum to constant ( cos(" n ) ) ( ) = cos(" n# )! Can also modify individual frames of and reconstruct! complex, time-varying modifications! tapered overlap makes things OK 38

20 STFT Analysis-Synthesis! e.g. oise reduction: STFT of original speech Speech corrupted by white noise Energy threshold mask 39 M